Multiple Dedekind zeta functions and evaluations of extended multiple zeta values

We define the number field analog of the zeta function of d-complex variables studied by Zagier in (First European Congress of Mathematics, vol. II (Paris, 1992), Progress in Mathematics, vol. 120, Birkhauser, Basel, 1994, pp. 497-512). We prove that in certain cases this function has a meromorphic continuation to C^d, and we identify the linear subvarieties comprising its singularities. We use our approach to meromorphic continuation to prove that there exist infinitely many values of these functions at regular points in their extended domains which can be expressed as a rational linear combination of values of the Dedekind zeta function.